New chaotic attractors: Application of fractal‐fractional differentiation and integration

Abstract

Very recently, the concept of fractal differentiation and fractional differentiation has been combined to produce new differentiation operators. The new operators were constructed using three different kernels, namely, power law, exponential decay, and the generalized Mittag‐Leffler function. The new operators have two parameters: the first is considered as fractional order and the second as fractal dimension. In this work, we applied these new operators to model some chaotic attractors, and the models were solved numerically using a new and very efficient numerical scheme. We presented numerical simulations for some specific fractional order and fractal dimension. The classical fractional differential models could be recovered when the fractal dimension is equal to 1; in these cases, the obtained attractors with power law presented no similarities. Nevertheless, those obtained via Caputo‐Fabrizio and the Atangana‐Baleanu derivative show some crossover effects, which is due to non‐index law property. However, those obtained from fractal‐fractional, in particular, those with the Mittag‐Leffler kernel, show very strange and new attractors with self‐similarities; these results are obtained for the first time. We conclude that this new concept is the future to modelling complexities with self‐similarities.

A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

“…In fact, having an overview of the on-going contributions to the theory and applications of fractional calculus, which are continually appearing in some of the leading journals devoted to mathematical and physical sciences, biological sciences, statistical sciences, engineering sciences, and so on, the subject-matter, which we have dealt with in this review article, is remarkably important and potentially useful. Moreover, the interested future researchers will surely benefit from the listing of references to some of the other applications of various fractional-calculus operators in the mathematical and other sciences, which we have not considered in the preceding sections (see, for example, [94][95][96][97][98][99][100][101][102][103][104][105][106][107][108][109][110][111]).…”

Sensitivity analysis of COVID-19 with quarantine and vaccination: A fractal-fractional model

“…The fractal-fractional (FF) operator, which combines fractal and fractional differentiation, is a novel idea whose characteristics and features are being researched right now. Fractal-fractional differentiation may be found in chaotic attractors, chemical processes, complex dynamical systems, and electrical circuits [33] , [34] , [35] , [36] , [37] .…”

Numerical solution of a fractal-fractional order chaotic circuit system

“…The fractal order deriva-tive and the fractional-order, as well as their combinations for the solution of physical problems have been suggested in Refs. [21][22][23][24][25][26]. The chaotic Shinriki's oscillator fractional model has been investigated by the authors in Ref.…”

“…[23,24]. The fractal-fractional operators have been applied effectively for the related problems, see [25][26][27], where the authors obtain results for new chaotic attractors, the study of the Hepatitis C model, and for system identification. Some more interesting work, where the authors proposed the fractional derivatives as an application to the scientific problems, see [28][29][30][31][32][33][34][35].…”